Download P.def.g.1. The first particle is in equilibrium.

DESCRIPTION OF REASONING PROCESSES
USED IN SOLVING PHYSICS PROBLEMS: REASONING
MAPS
Sevket GUNDUZ
Marmara University, Faculty of Education, Istanbul/TURKEY
Prof.Dr.Mehmet Ali CORLU
Marmara University,Faculty of Education, Istanbul/TURKEY
ABSTRACT
Problem solving requires a set of cognitive operations that is mainly reasoning. Solution can be obtained by applying these cognitive
processes using necessery strategies in compliance with the desired end or goal (Arık, 1987). Reasoning can be defined as a process of
deriving new information from the knowledge given explicitly guided by a rule (Kurtz,Gentre&Gunn, 1999). So, it can be said that the results or
judgements of each step in solution are obtained through reasoning by using the previously acquired judgements and physics rules, and these
derivation processes can be represented by graphical tools.
An analysis of reasoning process in problem solving leads the educational studies in the area of teaching problem solving and diagnosing the
source of students’ failures, because a student’s success in solving physics problems depends on reasoning processes that can be executed
successfully. These methods of analysis can also be utilized in the field of artificial intelligence enriching the expert systems’ databases (van
Someren, 1994), and developing the problem solving algorithms in softwares.
The aim of our study is to answer the following research questions: “How can we describe the reasoning processes employed in problem
solving explicitly?” and “How can we express the solution with graphical tools to make it easily understandable?” The main attempt is to portray
the methods developed as a reasoning map model. Thus, we will have valid, reliable and fast methods that serve our needs. New analysis
methods have been developed as an outcome of studies carried out with qualitative research techniques, The data was collected from the
interviews conducted with five experts and five students, and from the exam documents written by 153 students attending the 11th grade of a
public high school.
Preparation of the reasoning maps has been carried out in three stages: first, data have been collected from interviews, written documents and
observations; second, by means of the previously collected data, the knowledge elements have been determined, each of which acts as a rule
in reasoning, and the judgements produced in reasoning; third, the resoning processes have been defined with input/output judgements and
rules, and the relations between these processes. At the end of the third stage these descriptions have been shown in graphs. The processes
of constructing reasoning maps have been refined by analyzing data cyclically in a constant comparison method of data analysis (Lodico,
2006). This procedure validates data coming from similiar sources, Besides triangulation which is comparing data from different sources. In
this study, the process of preparing reasoning map model has been explained in detail, application to other fields has been left other studies.
Keywords: problem solving, reasoning, reasoning maps.
INTRODUCTION:
Problem solving is a process of finding the unknown
(Jonassen, 2000). Finding the unknown in general means
production of new information for the problem solvers. The
main function of problem solving is the reasoning process
(Arık, 1987). Reasoning is the mental process of deriving
logical consequences, which might take the form of a
judgement, conclusion or prediction, from given information
(Johnson-Laird, 2000; Simon, 1992; Kurtz et.al., 1999).
Therefore, there is a close relation between reasoning and
problem solving. It can be said that to analyze the reasoning
helps the analysis of problem solving directly. Restricting the
definition of problem within the physics makes the analysis
easy.
“Rule-based theories require that the rules are explicitly
represented and activated during processing; that they be
the causal force in the reasoning process” (Kurtz et.al.,1999,
p:153). Reasoning by rules has been applied most often to
the realm of propositional logic. As characterized by Rips,
(cited by Kurtz et.al., 1999) the premises are stored in
working memory,
then the application of rules is
coordinated by a reasoning program that searches for and
produces appropriate inferences by constructing and linking
steps in mental proof. This chaining process may proceed
forward and backward in a constrained way and may involve
building assertion trees in working memory. Human beings
have explicit mental inference rules that operate on and
transform propositions or judgements in working memory.
The rules specify the allowable ways in which information
can be put together, related, and structured.
Kurtz et.al.(1999) have defined the reasoning process in
functional form with three components, available information
x and k, cognitive processes brought to bear, and generated
inferences.
Y = F(x,k)
X represents the initial avaliable information, k refers to
stored knowledge about domain and particular cases used
in reasoning. F is a summary of the set of computational
tools used to manipulate, recombine, or transform the input
information; and y is the inferential product of reasoning
process.
We have defined the reasoning processes used in solving
physics problems similar to the Kurtz model. This new
model almost accounts for the reasonings for the physics
problems: however, in other areas, this model has not been
validated yet.
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The mental processes have a dynamic structure. Even if the
same outcome has been obtained in a problem solving, the
processes used in solution stages differ individually. Even if
the different people use the same processes, their use
orders and relations might vary.
rules
inputs
(x,k)
Operations
(evoking,
production, etc)
output
s (Y)
PROCESS
Figure 1. Structure of Reasoning
modelProcess
that is given in Figure-1 can
Our
be defined briefly
that initial avaliable information is used as an input, this
information evokes the rules from long term memory, and
judgements are produced as an output by the rules.
Judgements resulting from reasoning is a new information.
There are two main operations in the process: the first is
evoking the rules, and the second is production of actions or
results. Other psychological factors were neglected in this
model. Simon (1992) describes the productions in a
heuristic search system corresponding to the inference rules
in a system of logic. The general term “rule” that we have
used, as an inference rule, represents the Simon’s
production, conditional rule of modus ponens, and mental
models (Gentner, 2002; Markman&Gentner, 2001)’s.
Deductive, inductive and abductive thinking can be
represented in this model. Output judgements may create a
new appropriate situation, and they evoke the next
reasoning process. The outputs of the one process are
used as inputs of the next process. The chaining is
accomplished by the continually changing problem situation
that evokes the appropriate chain of reasoning. The
reasoning chains continue in accordance with the strategy
until obtaining the solution of the problem.
Initial avaliable knowledge that is given in physics problems
is mainly factual. It describes the situation, and it is
accepted to be true. The output information that is produced
by the input knowledge in reasoning process is inference or
derived information. The output information is a sentence of
judgement or conclusion that has a higher value than that of
the input knowledge. Output judgements can be divided in
some categories. It might be true, false or uncertain.
The rules that are used in processes can be categorized like
that principle, relation, law, model and theory. The relation is
a rule that is valid only for given problem, principle is a rule
that is valid for a problem group, i.e. more general than
relation. The range of generalization increases as the
authorized field of rules enlarges. The definitions are also
used as a rule in processes, e.g. let’s consider this question:
“what is the work done by force F, if it causes an object to
move its own direction in the distance x?” This question can
be answered with the definition of work as a rule. The
sentences which express generalization such as definition
and principles are called “knowledge element”. Knowledge
elements are also named as field knowledge (Bloom, 1982;
Özçelik, 1881). These elements are the basic factors
effecting the the problem solving performance (Fisher, 2000;
Kulm, 1994).
In this developed model, three methods of reasoning can be
graphically represented. In the first one, a judgement can be
raised, if the inputs and rules are known. This process is
called deduction – if there is no doubt on correctness of the
premises, no suspicion on the judgement as well. In the
second one, the rule can be set if the inputs and judgements
are known. This is what is called induction. As the number
of the processes repeated increases in deductive approach,
the reliability of the knowledge produced also gets higher.
This process is a type of generalization. In the third, though,
the input data can be found if the rule and judgement are
known. And this is called as ubduction – the recerse
application of deduction.
There are some cognitive events that can not be expressed
in the model, e.g. The determination of a loading question,
which affects the process selection after the solution starts
can not be expressed in this representation. However it can
be said that all questions are tried to be used which are
effective on knowledge production that will be helpful in
realization of the purpose related to research question
leading us to final result. This procedure is, in fact, an
inquiry. For this reason, the development of the inquiry
abilities of the students, which are effective on problem
solving will make them good problem solver. The inquiry
process or fact in prblem solving is a matter that is worth to
study on seperately.
The representation of a knowledge takes place an important
place in this model. Reasoning maps are graphical tools to
represent the reasoning processes employed in solution.
Jonessen (2003) argue that problem representation is the
key to problem solving. He describes the tools for
externalizing problem space, such as semantic networks for
modeling conceptual knowledge, expert systems for
representing procedural knowledge, systems modeling for
strategic knowledge. Semantic networks, also known as
concept maps or cognitive maps, are spatial representations
of concepts and their interrelationships that are intended to
represent the knowledge structuresthat humans store in
their minds (Jonassen, Beissner, & Yacci, 1993). Our
representation is different from those in that it represents the
reasoning processes employed in problem solving.
Purpose of the Study
This study aims to develop new model for the analysis of
reasoning employed in physics problem solving more
explicitly. In order to fulfill this aim, answers to the following
questions have been discussed:
1.How can we describe the reasoning processes employed
in problem solving explicitly?
2.How can we express the solution with graphical tools to
make it easily understandable?”
The main attempt is to illlustrate (portray) the methods
developed as a reasoning map model that the details has
given in findings. Thus, we will have valid, reliable and fast
methods that serve our needs.
METHOD
New analysis methods have been developed as an outcome
of studies carried out with qualitative research techniques
within the frame of interpretive paradigm (Cohen et.al.2000).
The research model is the grounded theory based on the
work of Glaser and Strauss (Lodico et.al. 2006). The data
was collected from the interviews conducted with five
experts and five students, and from the exam documents
written by 153 students attending the 11th grade of a public
high school.
Preparation of the reasoning maps has been carried out in
three stages: at first, the data from the solution of problem
given in Figure-2 has been collected from interviews, written
documents and observations; second, by means of the
previously collected data, the knowledge elements (Merill,
1999) have been determined, each of which acts as a rule in
reasoning, and the judgements produced in reasoning; third,
the reasoning processes have been defined with
input/output judgements and rules, and the relations
between these processes. At the end of the third stage
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these descriptions have been shown in graphs. The
processes of constructing reasoning maps have been
refined by analyzing data cyclically in a constant comparison
method of data analysis (Lodico et.al. 2006). This procedure
validates data coming from similiar sources, beside
triangulation which is comparing data from different sources.
In this study, the process of preparing reasoning map model
has been explained in detail, while application to other
fields has been left for other studies.
Table 1. The list of theme defined by content analysis of the
problem solution.
Codes
p.def…
Judgements regarding the definition the initial
givens in problem case.
p.gen... Judgements about determining the principles
and laws of problem case.
p.equ… Judgements regarding the application of first
principle of equilibrium.
p.dyn... Judgements regarding the application of
Newton’s seconed law.
p.cou... Judgements about defining the interactions of
charged particles according to Coulomb’s law.
p.efi...
Judgements about defining the interaction of
charged particle with electric field.
p.tef...
Judgements about finding the resultant electric
field for a given point.
p.sol...
Judgements regarding the application of existing
principles and laws in problem.
FINDINGS AND INTERPRETATION
Data coming from the sources have been evaluated and the
reasoning map model has been developed. In this section
the last version of developed model has been explained,
and findings about the initial stages have been mentioned.
Some definitions and rules of the model concluded by
analysis of the data are given below, and application of the
model evaluated through one individual’s solution.
Findings About the First Stages
As a subtopic of physics, “electrostatic force and field” was
selected. 21 knowledge elements belonging to this topic
were determined and these were grouped in categories of
Coulomb force, electric field and equilibrium state. The last
one was added after the preparation of problem case. The
elements have been coded with the number of two digits,
the first shows the category, the second shows the
sequence. For example, “3.i. The force exerted on
negatively charged particle placed in an external electrical
field has a direction opposite to field direction”. Codes in
Figure-3,4,5 that are enclosed by circles show the
knowledge elements which act as a rule to direct the
reasoning process. Then, the problem case and
fundemental questions have been determined as given in
Fig-2. The problem was solved by 5 physics teachers in
interviews.
48 judgements were found from the teachers’ interviews,
these are sentences giving information about the problem
solution and produced in each solution step as a product of
reasoning processes. Judgements have been coded like
“p.equ.a”, “p.cou.b”,…symbols. The first character, “p”
means of product of process. Second group of characters,
“equ”, “cou”, “efi”,…etc. means the theme of judgements
out of eight groups given in Table-1. For example “equ” is
about equilibrium, “cou” is about Coulomb force, “efi” is
about electric field. The third group of characters indicates
the sequence of judgements, i.e. a is the first, b is the
second, etc. A few judgements were needed to be put into
fourth category, in that case a new character set was used
like “x”, “y”, and “z”. For example “p.equ.b.x” from the
Figure-6 means that this is the first part of the second
judgement produced as an outcome of reasoning process
directed by the rules of vector addition. The judgement is
“The force Fe due to interaction of electric field is equal to
the force Fc due to mutual interaction i.e. Coulomb Law”.
The coding symbols were chosen arbitrarily.
Name of theme
Reasoning map given in Figure-6 has been prepared after
the judgements were determined. That shows the reasoning
processes used in solution and relations of each process.
Each process consists of knowledge elements (enclosed by
circles in figures) and input-output judgements (encloesd by
squares). The direction of arrow shows knowledge process
from input to output.
Findings Regarding the Definitions of the Model
Some definitions given here are new and some of them are
common general terms. We have prefered to mention them
again here to define the model as a whole.
Definition-1. Process/subprocess: A process can be broken
down into three basic components: the inputs, outputs, and
the rules governing the operations transforming the inputs to
outputs. Every process can be defined in terms of
subprocess.
Definition-2. Unit/fundemental/core process: is the
definable smallest process. It can not be decomposed into
other processes. It is usually governed by one rule.
Definition-3. Process data and judgement: Input of the
process is called datum, output of the process is called
judgement.
Definition -4. Factual Knowledge: is knowledge that is not
product of infering, but given about the problem case or
defining the problem case.
In order to clarify the meaning of the codes used in
reasoning maps, description of the categories are provided
below:
p.dyn.d.1 This code indicates that it is the first value of the
judgement “d” belongs to “dyn” theme of “p” process.
“.dyn.“ indicates theme of the application of Newton’s
seconed law.,
“.d.“ indicates the judgement of “ if the external electric field
increases, then magnitude of net force increases outward, if
field decreases, net force increases inwards.
“.1.“ indicates the specific judgement “if the external electric
field increases, then magnitude of net force on the point of
the first particle increases to the left direction”.
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the scientific definition or principle. These are active in
problem solving, and without knowing them it is impossible
to carry out the solution (Fiser, 2000; Kulm, 1994; Özçelik,
1981).
Problem Case:
Findings Regarding the Induced Principles of
Model (Some obtained generalizations):
Left
T1
T2
d
First particle
m1, q1
right
Low level generalizations, or induction of the rules, or
induced principles have been drawn from the findings.
These principles have been used in the analysis of the
problem solution, and they were considered to be valid in
the set of facts that the model covers. The induced
principles about the model are explained below.
Second Particle
m2, q2
Two point charged particles hanging vertically by
strings in a horizontal electric field are in equilibrium.
Fundemental Questions:
Answer the following questions by using the problem
case;
a) What is the ratio of charge values, q1 /q2 =?
b) What is the magnitude of external electric field in
terms of givens?
c) What is the direction of external electric field and the
signs of charges?
d) How do the positions of particle change, if the
magnitude of external electric field and charges of
point materials, and distance of materials change?
Figure 2. Problem case and related fundamental
questions ask in interviews.
Definition-5. Process rule: is any rule governing the
operations related with inputs and outputs (judgements) in
reasoning.
Definition-6. Verified Judgement: If the data and rule of
process are exactly known as correct or verified that they
are true, the output is called verified judgement.
Definition-7. Unverified Judgement: If accuracy of either the
data or the rule of process has not been verified yet, the
output is called unverified judgement.
Definition-8. Directing question and aim of process:
Directing question causes the process to start, and clarify
the purpose of process. Purpose of the process are
embedded in question implicitly. Operations are governed
according to purpose.
Definition-9.Unifold and manifold process: The process that
produce single judgement is called unifold, if there are more
than one judgement it is called manifold process.
Definition-10. The process name: is the name of inference,
and determined according to the judgement infered.
Definition-11. Range of generalization (Induction, drawing a
rule): If the input and the output of the process are known,
then the rule of process can be drawn inductively. This is
called generalization or induction. The knowledge obtained
from generalization within the solution is called as byprinciples.
Definition-12. Reasoning maps: The graphs showing the
reasoning process with the input, the output, the rules, and
the relation between them are called reasoning maps.
Definition-13. Category of the process: The process is
categorized according to type of judgement.
Definition -14. Flowing direction of Knowledge: The
direction of processing the knowledge is towards from the
data to judgement
Definition- 15. Evoking : is an operation that the data
evokes the process rule from the long term memory (Simon,
1992).
Definition- 16. Production: is an operation that the
judgement is produced after evoking (Simon, 1992).
Definition- 17. Knowledge Element: The knowledge
elements (Merrill, 1999) can be regarded as content
knowledge (Bloom, 1982). A knowledge element represents
Principle-1. The general process that define the solution of
the physics problem consists of the sequence of unit
processes related to each other.
Principle-2. The unit processes trigger each other in chain
reactions. The output of one process causes the related
processes.
Principle-3. The output information has a higher value than
the input information, because the output is produced as a
consequence of cognitive operations.
Principle -4. Unit process is usually unifold, and rarely
twofold.
Principle -5. The well defined inputs makes it easy to find
the rules from long term memory. Sometimes, after the the
rules have been found they help the inputs to be well
defined, if they are not at the beginning. There are two way
interaction between input and rules.
Principle -6. This model explains the structure of inquiry.
Principle-7. Deductive reasoning can be represented by
this model.
Principle-8. Inductive reasoning can be represented by this
model.
Principle-9. While the generalization level of produced
knowledge becomes high, the area of validity becomes
wider.
Principle-10. If knowledge has been produced by
generalization in one proces, it can be used as a rule that is
called by-principle in other process. So, by-principles can be
used in reasoning process.
Principle-11. The reasoning maps of problem solvers are
different even if they solve the same problem.
Principle-12. If the number of processe for the problem
increases, the difficulty of a problem increases. Difficulty of
a problem is proportional with the number of processes.
Principle-13. If the number of process for the problem
increases, the need for the mathematical knowledge, that is
probability of using math increases.
Principle-14. The mean of difficulty index of the questions
measuring the outputs is lower than the mean of questions
measuring the inputs.
Principle-15. There are high correlations between the
scores of the questions measuring the inputs and outputs.
Principle-16. Correlations between the scores of the
questions measuring the inputs may not be high.
Principle-17. Correlations between the scores of the
questions measuring the outputs may not be high
Principle-18. The success of a processes can be assesed
by using the scores of question measuring the outputs
Principle-19. Johnson-Laird’s mental model can be
represented by reasoning maps. The elements of process
may be mental models.
Principle-20. Gentner(2001)’s causal mental model can be
represented by reasoning maps.
Principle-21. Production of the output or judgement
requires the ability of application.
Principle-22. The judgements may be in the types of
analysis, synthesis, an evaluation.
Principle-23. Heuristic reasoning can be represented by
reasoning maps.
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Principle-24. The strategy used for the solution can be
determined from the reasoning maps.
Principle-25. This model doesn’t explain the seeing.
Findings Regarding the Application of Model
Here is an example of the application of the model in
analysis of solution of one interviewee. Solver has begun
the solution by searching the content area.
“… since the charged particles exist, problem is related with
static electric, …related with electrical force. …there are
particles in equilibrium. .. related with equilibrium….”
These words indicate (point) to principle-5. It means that
givens in the problem evoke the Coulomb’s Law knowledge
Coulomb’s Law
is active.
Particles
are charged
Rule 2.b: Two
Charged particles
apply a force upon
each other.
Coulomb
force Fc are
applied on
each particle
Outputs :
1. Coulomb force Fc is
applied on first particle
2. Coulomb force Fc is
applied on second particle
Figure 3. Open diagram of unit process.
The level of validity or influence of the process’ judgement
works only for this problem, and it can be used as an input
in all processes if necessery.
P.cou.a
P.def.a
Outputs :
P.cou.a. Coulomb force Fc are applied
on each particle
P.cou.a.1. Coulomb force Fc is applied
on first particle.
P.cou.a.2. Coulomb force Fc is applied
on second particle.
This process can not be divided into small process
practically, so this can be considered as a unit process (D-2).
The direction of knowledge flow in this process is
bidirectional. It means, if it is known through other ways that
the electrical force is exerted upon particles it may be
infered that the particles are charged.
Figure-4 Coded diagram of the unit process.
schema in mind, and make the rules easy to be found from
it. The first reasoning that the solver used has been given
below:
“1-… [p.gen.b] Charged particles apply a force on
each other, because of Coulomb’s Law. [p.def.a] The first
and second particles given in the problem are charged, so
[p.cou.a] these two particles exert a force upon each other.”
The first two judgements have been produced by seeing
differently from the formal reasoning. The third can be used
as the name of process. This reasoning process has been
represented in figure-3 graphically. The components of the
process have been given in Table-2 in accordance with the
developed model.
Table 2. Components of “P.cou.a” process
Components
Contents
The inputs of the process are factual knowledge (definition-4)
and are used as data (def-3). They are not produced in
reasoning, but seeing which may be regarded as perception
process. Seeing process can not be explained by this
process (principle-25). The output of process is a judgement
(D-3) produced by applying the process rule (D-5). This is
called production (D-16). The interviewee did not do
anything to verify that process rule is correct or not. He was
certain that it was a true and well-known principle.
This reasoning process is deductive. Since major premise is
not given in the problem, and it is used from the cognitive
schema of physics knowledge, so it may be regarded as
heuristic reasoning (p-23). There are cause and effect in this
process, so it may be named as causal reasoning.
P.gen.b
inputs:
P.def.a. Particles are charged, q1 and q2
p.def.a.1. First particle is charged q1.
p.def.a.2. Second particle is charged q2.
P.gen.b. Coulomb’s Law is active in problem,
Rule :
2.b: Electrical charged particles exert a force
on each other.
This process may be used in other problems where there
are charged particles. The detailed analysis of process in
terms of model is given below.
The process’ aim and the directing question aren’t
expressed in the diagram. Since the aim of the process is to
produce the output, the output implies the aim and the
directing question implicitly.
Inputs:
1. First particle is charged q1.
2. Second particle is charged q2.
3. Coulomb’s Law is active.
2.b
This is the process that it has produced p.cou.a output from
p.gen.b and p.def.a inputs by using the rule 2.b The
meaning of codes are given in table-2. This process is given
in coded schema in figure-4, where inputs, outputs and rule
are given with code numbers. This technique is prefered in
order to show more processes in per page.
In figure-4 the input and output are expressed the most
specific form like p.def.a.1, p.def.a.2, p.cou.a.1, p.cou.a.2.
These are the particle specific sentences. P.def.a, p.cou.a
are more general judgements than the particle-focused
sentences.
Finding the rules (principles) from the knowledge schema
affects the problem solving. The finding algorithms or
evoking ability which can be named remembering or
application skill varies from person to person.
Problem solution has been continued;
“2-… [p.def.f] The line connecting the particles is
horizontal, so [p.cou.d] the direcetion of force exerted on
particles is horizontal. ….”
The components of this infering are given in Table 3.
Tablo 3. Components of “P.cou.d” process
Process
name
P.cou.a.Infering process that particles exert
a force on each other because of their
charges.
Directing
How does an effect appear due to particles
question
to be charged?
İnputsp.def.a.1. First particle is charged q1.
(data) p.def.a.2. Second particle is charged q2.
p.gen.b. Coulomb’s Law is active in problem.
Process
2.b. Two charged particles apply a force on
rule
each other.
Outputs
P.cou.a.1. Coulomb force Fc is applied on first particle.
(judgement
P.cou.a.2. Coulomb force Fc is applied on second
s)
particle.
Components
Process name
Contents
P.cou.d. Infering process that the
dicetion of force exerted on particles
due to their charges is horizontal.
Directing
What is the direction of electrical forces
question
exerted on particles?
İnputs (data)
p.def.f. The line connecting the particles is
horizontal.
P.cou.a.1. Coulomb force Fc is applied on first
particle.
P.cou.a.2. Coulomb force Fc is applied on second
particle.
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particles. The questionthat was asked to the interviewee
was: “what did you think between the third and fourth
sencences?” He replied that:
Process rule
2.e. The direction of the Coulomb
Force exerted on each other is on the
line connecting particles charge q1 and
q2.
Outputsp.cou.d.1. The force Fc applied on the first particle
(judgements)
is horizontal.
p.cou.d.2. The force Fc applied on the first particle
is horizontal.
The output of the process “p.cou.a” was used here as the
input of the process “p.cou.d” (D-3). If the sequence is
considered, p.cou.a must precede
p.cou.d. The first
process has triggered the second. These two processes and
their relations have been shown in figure-5.
This model briefly represents the scientific method. In
scientific method, if the factual data (inputs) are known, the
principles (rules) are constructed theoretically from them,
this is called invention. The hypothesis can be infered by
using the constructed frame of theory. If the validity of
hypothesis are verified either theoretically or experimentally,
the validity of the principles are also verified. These
procedures can be represented by reasoning maps.
As the analysis of problem solving is continued:
“3-… Since the objects are in equilibrium, the net
forces exerted upon the objects are equal to zero. …
[p.equ.a]”
The components of this process are as follows:
Input: P.def.g. Particles are in equilibrium.
P.def.g.1. The first particle is in equilibrium.
P.def.g.2. The second particle is in equilibrium.
Rule: 1.a.If an object is in equilibrium, then the resultant
force exerted upon that particle is equal to zero
Output: P.equ.a. the resultant forces exerted upon the
particles are equal to zero.
P.equ.a.1. the resultant force exerted upon the first particle
is equal to zero.
P.equ.a.2. the resultant force exerted upon the second
particle is equal to zero.
This inference has been made according to the first principle
of equilibrium, that is special case of Newton’s First Law.
This process has been triggerd by the electrical force
obtained previously and initial information of being
equilibrium given in the probem case.
“4-… since the objects are in the horizontal homogenous
external electric field, a secondory electrical force is exerted
upon the objects due to that field. And that force is
horizontal. …” [p.efi.a]
The components of this process are as follows:
Inputs:
P.def.e. There exists a homogeneous external electric field.
P.def.c. The direction of external electrical field is horizontal.
P.gen.c. Charged particles interact with electrical field.
Rule: 3.a.The electrical force is exerted upon charged
objects placed in an external electrical field in the direction
of field.
Output: P.efi.a. The horizontal electrical force Fe is exerted
upon the particles due to external electrical field.
P.efi.a.x.1. Fe1 is exerted upon first particle.
P. efi.a.x.2. Fe2 is exerted upon second particle.
P. efi.a.y.1. Fe1 is horizontal.
P. efi.a.y.2. Fe2 is horizontal.
The judgement that the net forces acted on particles are
zero implies that there must be a second force applied on
“5-… in fact I found that the net force was zero, but,
because of vector addition if the net force is zero there must
be more than one force. I have looked to weights and
tensions, these are perpendicular to the electrical forces
[existing due to the charges of particles], I couldn’t use
them. And again I looked at the problem and I realized the
external electrical force. I thought to consider the force due
to that field….”
Interviewee executed the process of finding the resultant
vector (p.equ.b) after the process of p.equ.a. Since the data
was insufficient, he couldn’t succeed (principle-5). The
evoked rule has been implied that there must be an
additional data, this implication caused the interviewee to
ask new questions and to produce the process “p.efi.a”
explained above. This is in fact the process of inquiry.
Interviewee executed the process p.equ.b:
“6-… The [horizontal] forces acted on objects equal
each other and oppsite direction. ..”(p.equ.b)
The analysis of individual’s solution can be proceeded by
using this method as explained in model. Our purpose is to
explain the application of our model in the analysis of the
physics problem solution, so we didn’t mentioned any more
details. The reasoning map of the first question’s solution is
given in the Figure-6. In short, it can be said from the map
that this map varies individually, and gives the information
about the solution strategies of the persons. At the stage of
physical description thare are 5 processes, at the stage of
converting the physical representation to mathematical
representation there are 4 processes, and finall there is one
process at the stage of equation solution.
CONCLUSION AND SUGGESTIONS
The main purpose of this study is to describe the model
explaining explicitly the resoning processes employed in
problem solving. First of all, the definitions and principles of
the model that was constructed by the qualitative analysis of
the solutions of the participants were given, then the
individual’s solution was explained by using the model. This
model has of course some limitations. It is hard to represent
the higher order thinking process.
Having simple explanations of processes facilitates the
assesment of problem solving. Application of the model to
the assesment of problem solving has been made already in
another study (Gunduz, 2007). Gunduz has shown how the
resoning maps can be used in preparing the diagnostic test
of physics. Judgements given with a reasoning map indicate
the objectives that the items measure. Questions measuring
the judgements have been prepared accordingly for the
diagnostic test.
In addition to these studies, it is recommended to search
how the reasoning maps can be used in teaching problem
solving. Jonessen (2003) argue that problem representation
is the key to problem solving. He mentions in his article that
Mayer reports that diagrams or flowcharts produces better
performance than verbal representations, especially for
more complex problems. He also cites that the spatial
reorganization of information facilitates some of the
cognitive activities that are required to solve problems.
ACKNOWLEDGEMENT: This study was supported by the
Scientific Research Projects Department of Marmara
University with the project numbered EĞT-117/081004 and
dated 08.10.2004.
1017
Question:what are the relations between the
magnitude of charges, external electrical field and
distance between particles?
Particles are
in equilbrium
The line connecting
particles is
horizontal
Particles are in the
uniform electrical
field Ed.
Paricles are
charged
Ed is
horizontal
Level of determining
the givens
1th rule of
equilibrium
(1.a)
E Field
(3.a)
Coulomb’
s Law
(2.b)
Fnet=0
Stages of physical
description: Determination of
definitions/principles/laws.
Fc is
horizontal
2.e
2.d
Fc
is acted
Fc1=Fc2
Horizontal Fe
is acted
2.c
4.a
3.g
FC 
Fc and Fe are
horizontal and
opposite.
Fe=Fc
kq1q2
d2
the stage of converting the physical
representation to mathematical
representation: Aplication of rules and
writing the equations
Math rule
(5.a)
If an object is in equilibrium, then
the resultant force exerted upon that
particle is equal to zero. Particles
are in equilibrium. So, the resultant
forces exerted upon the particles are
equal to zero.
q1Edis 
kq1q2
q2 Edis 
d2
particle-1
Fe = q.E
kq1q2
d2
particle -2
Math rule
(5.a)
Edis 
kq2
d
2

kq1
q1 = q2
d2
Figure 6. Reasoning Map of solution given to the first question of problem case.
1018
The stage of the
equations’ solution
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